Method for blind demodulation at higher orders of several linear waveform transmitters

ABSTRACT

Disclosed is a method of blind demodulation of signals arising from one or more transmitters, the signals including a mixture of symbols comprising at least one step of separating the transmitters by using the temporal independence of the symbol trains specific to a transmitter and the mutual independence of the transmitters.

The invention relates notably to a method of blind demodulation at higher orders of linear waveform where the signals of several radiocommunication transmitters are received on a system of several antennas.

Antenna processing processes the observations originating from several sensors.

FIG. 1 shows a system of antennas composed of an array with several antennas receiving several radio-electric sources with different angles of incidence. The antennas of the array receive the sources with a phase and an amplitude dependent on the angle of incidence of the sources, as well as the position of the antennas. FIG. 2 shows that the angles of incidence of the sources can be parametrized either in one dimension, 1D, with the azimuth θ_(m) or in two dimensions, 2D, with the angles of azimuth θ_(m) and elevation Δ_(m).

Antenna processing techniques utilize the spatial diversity of the sources: use of the spatial position of the antennas of the array so as to better utilize the differences in incidence and in distance of the sources. Antenna processing breaks up into two major areas of activity:

Goniometry, the objective of which is to determine the incidences θ_(m) in 1D or the pair of incidences (θ_(m), Δ_(m)) in 2D. For this purpose, goniometry algorithms use the observations arising from the antennas or sensors. FIG. 2 shows that goniometry is performed in one dimension, 1D, when the waves from the transmitters propagate in the same plane and that otherwise it is necessary to apply goniometry in two dimensions, 2D. This plane of the waves is often that of the antenna array where the angle of elevation is zero.

Spatial filtering, illustrated in FIG. 3, the objective of which is to extract either the modulated signals s_(m)(t), or the symbols contained in the signal (Demodulation). This filtering consists in combining the signals received on the sensor array so as to form an optimal reception antenna for one of the sources. Spatial filtering can be blind or cooperative. It is cooperative when there exists a priori knowledge about the signals transmitted (direction of arrival, symbol sequences, etc.) and it is blind in the converse case. Included in this activity are the activities of blind separation of sources, matched filtering on direction of arrival (beamforming) or on replicas, multi-sensor MODEM (demodulation), etc.

The current techniques of multiple input multiple output or MIMO blind demodulation [11] [12][13][14], have notably the drawback of processing only the case of baseband transmitters with 1 sample per symbol. In these techniques, there exist procedures utilizing solely statistics of order 2 [12]. Other procedures are extensions of the CMA technique [11] which, in particular, in single input multiple output or SIMO, have the drawback of converging less empty than order-2 procedures [5] [9]. The procedure in [13] has notably the drawback of demodulating the transmitters one after another by an iterative technique of successive elimination of the transmitters to be demodulated. This approach exhibits the drawback of not processing the transmitters in an equal manner.

The invention relates to a method of blind demodulation of signals arising from one or more transmitters, the signals consisting of a mixture of symbols where the signals are received on a system comprising several receivers characterized in that it comprises at least one step of separating the transmitters by using the temporal independence of the symbol trains {a_(k-p,i)} indexed by “p” specific to a transmitter and the mutual independence of the transmitters, being the index of a transmitter by “i”.

The method according to the invention exhibits notably the following advantages:

-   -   the symbol rates of the transmitters can be different and are,         consequently, not necessarily equal to 1 sample per symbol,     -   the transmitters are not necessarily baseband and can have         different carrier frequencies,     -   the transmitters are demodulated jointly without performing a         technique of iterative demodulation of each of the transmitters.         The technique does not make any assumption about the         constellation as in document [11],     -   the transmitters can have different shaping filters,     -   the method is not affected by an over-estimation of order of the         model as in [12] involving ARMA models (Auto Regressive with         Adapted Mean)

Other characteristics and advantages of the present invention will be better apparent on reading the description which follows of an exemplary embodiment given by way of wholly nonlimiting illustration accompanied by the figures which represent:

FIG. 1 a diagram comprising transmitters and an antenna processing system,

FIG. 2 the representation of an incidence of a source,

FIG. 3 spatial filtering by beamforming in a direction,

FIG. 4 a schematic of the demodulation of the symbols of the m^(th) transmitter in the MIMO context,

FIG. 5 a transmitter with linear modulation,

FIG. 6 an exemplary constellation of a phase-shifted 8-QAM modulation,

FIG. 7 a schematic of the steps of a first variant embodiment of the invention, and

FIG. 8 a diagram of the steps of a second variant of the invention.

The method according to the invention relates notably to the demodulation, that is to say the extraction of the symbols {a_(km)} transmitted by the m^(th) transmitter.

FIG. 4 illustrates the propagation of a signal through a multi-path channel. The m^(th) transmitter transmits the symbol a_(k,m) at the instant kT_(m) where T_(m) is the symbol period. Demodulation consists in estimating and detecting the symbols so as to obtain the symbols â_(km) at the output of the demodulator. FIG. 4 shows the case of two transmitters with linear modulation: the symbol train {a_(k,m)} is filtered linearly by a transmission filter also called the shaping filter. The transmission filters of each of the transmitters may be different.

The method is concerned notably with techniques of blind demodulation of the symbols {a_(k,m)} of several transmitters indexed by “m” with linear modulation. Blind techniques do not use any a priori information about the signals transmitted: shaping filter, training sequence, etc.

Before making explicit the steps implemented by the invention a few reminders necessary for the understanding thereof are given.

Linear Modulation

The diagram of FIG. 5 shows the process of the linear modulation of a symbol train {a_(k)} with the rate T by a shaping filter h₀(t).

The symbol comb c(t) is first of all filtered by the shaping filter h₀(t) and thereafter transposed to the carrier frequency f₀. The NRZ filter, which is a temporal gate of length T and defined by h₀(t)=Π_(T)(t−T/2), is a particular example of a transmission filter. In radiocommunications, use is also made of the Nyquist filter whose Fourier transform h₀(f)≈Π_(B)(f−B/2) approximates to a gate of band B (the roll-off defines the slope of the filter outside of the band B, when the roll-off is zero then h₀(f)=Π_(B)(f−B/2)). The modulated signal s₀(t) may be written at the instant t_(k)=kT_(e) (T_(e): sampling period) as a function of the symbol comb c(t) in the following manner:

$\begin{matrix} {{s_{0}\left( {kT}_{e} \right)} = {\sum\limits_{i}{{h_{0}\left( {iT}_{e} \right)}{{c\left( {\left( {k - i} \right)T_{e}} \right)}.}}}} & (1) \end{matrix}$

We take for example a symbol time equal to an integer number of times the sampling period, in other words, we put T=IT_(e) and -therefore, k=mI+j with 0≦j<I. Since c(t)=Σ_(r) a_(r) δ(t−rIT_(e)), stated otherwise, as c(t)=a_(u) for t=uIT_(e) and c(t)=0 for t≠iIT_(e), the only values of i for which c((k−i)T_(e)) is nonzero satisfy k−i=uI, that is to say such that i=mI+j−uI=nI+j where n=m−u. Ultimately, the expression (1) becomes:

$\begin{matrix} {\begin{matrix} {{s_{0}\left( {{mIT}_{e} + {jT}_{e}} \right)} = {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{0}\left( {{nIT}_{e} + {jT}_{e}} \right)}a_{m - n}}}} & {{{for}\mspace{14mu} 0} \leq j < I} \end{matrix}.} & (2) \end{matrix}$

The parameter L₀ is the half-length of the transmission filter which is spread over a duration of (2L₀+1)IT_(e). In the particular case of an NRZ transmission filter, we obtain L₀=0. As regards the signal s(t), it satisfies s(t)=s₀(t)exp(j2πf₀t), since it is equal to the signal s₀(t) transposed to the frequency f₀. Under these conditions, the expression for s(mIT_(e)+jT_(e)) is the following according to (2):

$\begin{matrix} {\quad\begin{matrix} {{s\left( {{mIT}_{e} + {jT}_{e}} \right)} = {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{0}\left( {{nIT}_{e} + {jT}_{e}} \right)}{\exp \left( {j\; 2\pi \; {f_{0}\left( {{nI} + j} \right)}T_{\underset{\_}{e}}} \right)}a_{m - n}}}} \\ {{{\exp \left( {j\; 2\pi \; {f_{0}\left( {m - n} \right)}{IT}_{e}} \right)}.}} \\ {= {\sum\limits_{n = {- L_{0}}}^{L_{0}}{{h_{F\; 0}\left( {{nIT}_{e} + {jT}_{e}} \right)}b_{m - n}}}} \end{matrix}} & (3) \\ {{{such}\mspace{14mu} {that}\mspace{14mu} 0} \leq j < {I.}} & \; \end{matrix}$

where h_(F0)(iT_(e))=h(iT_(e))exp(j2πf₀iT_(e)) and b_(i)=a_(i) exp(j2πf₀iIT_(e))

Reception of the Signals on the Sensors: MIMO Case

The Multiple Input Multiple Output, or MIMO for short, model is a system composed of N>1 antennas (MO) which receives a mixture of several linear modulation transmitters with signal s_(i)(t) (MI) and symbol time T_(i). More particularly, the signals s_(i)(t) of each of the transmitters are linear modulations with I_(i) samples per symbol, of waveform h_(i)(t) and carrier frequency f_(i) such that:

$\begin{matrix} \begin{matrix} {{s_{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e}} \right)} = {\sum\limits_{n = {- {Lq}}}^{L_{q}}{{h_{q}\left( {{{nI}_{i}T_{e}} + {jT}_{e}} \right)}{\exp \left( {j\; 2\pi \; {f_{i}\left( {{nI}_{i} + j} \right)}T_{\underset{\_}{e}}} \right)}}}} \\ {{a_{{m - n},q}{\exp \left( {j\; 2\pi \; {f_{i}\left( {m - n} \right)}{IT}_{e}} \right)}}} \\ {= {\sum\limits_{n = {- L_{q}}}^{L_{q}}{{h_{Fi}\left( {{{nI}_{i}T_{e}} + {jT}_{e}} \right)}b_{{m - n},i}}}} \end{matrix} & (4) \\ {{{such}\mspace{14mu} {that}\mspace{14mu} 0} \leq j < {I.}} & \; \end{matrix}$

where h_(Fi)(kT_(e))=h_(i)(kT_(e)) exp(j2πf_(i)kT_(e)) and b_(k,i)=a_(k,i) exp(j2πf_(i)kI_(i)T_(e)) where the {a_(k,i)} are the symbols transmitted by the i^(th) transmitter and L_(i) is the half-length of the transmission filter of the i^(th) transmitter.

FIG. 4 shows that the signal s_(i)(t) of the i^(th) transmitter passes through a propagation channel before being received on an array composed of N antennas. The propagation channel can be modeled by P_(i) multi-paths of incidence θ_(pi), delay ι_(pi) and amplitude ρ_(pi) (1≦p≦P_(i)). At the output of the sensors, the M signals s_(i)(t) are received on the sensors and the vector x(t) is the sum of a linear mixture of the P_(i) multi-paths of each of the M transmitters. This vector of dimension N×1 has the following expression:

$\begin{matrix} \begin{matrix} {{{x(t)} = {{\sum\limits_{i = 1}^{M}{x^{i}\left( {t + t_{i}} \right)}} + {{b(t)}\mspace{14mu} {where}}}}\mspace{11mu}} \\ {{x^{i}(t)} = {{\sum\limits_{p = 1}^{P_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{s_{i}\left( {t - \tau_{pi}} \right)}}} = {A_{i}\Omega_{i}{{s_{i}(t)}.}}}} \end{matrix} & (5) \end{matrix}$

where ρ_(pi) is the amplitude of the p^(th) path of the i^(th) transmitter, s_(i)(t) is the signal of the i^(th) transmitter, b(t) is the noise vector assumed Gaussian, a(θ) is the response of the sensor array to a source of incidence θ, A_(i)=[a(θ_(1i)) . . . a(θ_(Pi))], Ω_(i)=diag([ρ_(1i) . . . ρ_(pi,i)]) and s_(i)(t)=[s_(i)(t−τ_(Pi,i)) . . . s_(i)(t−τ_(Pi,i))]^(T). Noting that τ_(pi)=r_(pi) I_(i) T_(e)+Δτ_(pi) where (0≦τ_(pi)<I_(i)T_(e)) and using expression (4) in equation (5), we obtain:

$\begin{matrix} {{.{x^{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e}} \right)}} = {\sum\limits_{p = 1}^{P_{i}}{\sum\limits_{n = {- L_{i}}}^{L_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{h_{Fi}\begin{pmatrix} {{{nI}_{i}T_{e}} + {jT}_{e} -} \\ {\Delta \; \tau_{pi}} \end{pmatrix}}{b_{{m - n - r_{p}},i}.}}}}} & (6) \end{matrix}$

By making the following change of variable u_(pi)=n+r_(pi), we obtain:

$\begin{matrix} {{.{x^{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e}} \right)}} = {\sum\limits_{p = 1}^{P_{i}}{\sum\limits_{u_{i} = {r_{pi} - L_{i}}}^{r_{pi} + L_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{h_{Fi}\begin{pmatrix} {{\left( {u_{pi} - r_{pi}} \right)I_{i}T_{e}} +} \\ {{jT}_{e} - {\Delta\tau}_{pi}} \end{pmatrix}}{b_{{m - u_{p}},i}.}}}}} & (7) \end{matrix}$

Now, putting r_(min,i)=min{r_(pi)} and r_(max,i)=max{r_(pi)}, the previous equation can be written in the following manner:

$\begin{matrix} {{.{x^{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e}} \right)}} = {\sum\limits_{p = 1}^{P_{i}}{\sum\limits_{u = {r_{\min,i} - L_{i}}}^{r_{\max,i} + L_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{h_{Fi}\begin{pmatrix} {{\left( {u - r_{pi}} \right)I_{i}T_{e}} +} \\ {{jT}_{e} - {\Delta\tau}_{pi}} \end{pmatrix}}{{Ind}_{\lbrack{{{rpi} - {Li}},{{rpi} + {Li}}}\rbrack}(u)}{b_{{m - u},i}.}}}}} & (8) \end{matrix}$

Where Ind_([r,q])(u) is the customary indicator function defined on the set of integers relating to value in the binary set {1, 0}, characterized by Ind_([r,q])(u)=1 if u belongs to the interval [r,q] and Ind_([r,q])(u)=0 otherwise. Therefore, denoting by v^(i)(t) the channel vector of the i^(th) transmitter:

$\begin{matrix} {{.{v^{i}\left( {{{uI}_{i}T_{e}} + {jT}_{e}} \right)}} = {\sum\limits_{p = 1}^{P_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{h_{Fi}\begin{pmatrix} {{\left( {u - r_{pi}} \right)I_{i}T_{e}} +} \\ {{jT}_{e} - {\Delta\tau}_{pi}} \end{pmatrix}}{{{Ind}_{\lbrack{{{rpi} - {Li}},{{rpi} + {Li}}}\rbrack}(u)}.}}}} & (9) \end{matrix}$

where t=u I_(i) T_(e)+jT_(e) and expression (6) becomes:

$\begin{matrix} {{{.{x^{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e} + t_{i}} \right)}} = {\sum\limits_{u = {r_{\min,i} - L_{i}}}^{r_{\max,i} + L_{i}}{{v^{i}\left( {{{uI}_{i}T_{e}} + {jT}_{e} + t_{i}} \right)}b_{{m - u},i}}}}{{{with}\mspace{14mu} 0} \leq j < {I_{i}.}}} & (10) \end{matrix}$

Denoting by I the greatest common multiple of the integers I_(i) (1≦i≦M) and J_(i) the integer satisfying I=I_(i) J_(i), equation (5) becomes at t=m I T_(e)+jT_(e):

$\begin{matrix} {{{x\left( {{mIT}_{e} + {jT}_{e}} \right)} = {{\sum\limits_{i = 1}^{M}{\sum\limits_{u = {r_{\min,i} - L_{i}}}^{r_{\max,i} + L_{i}}{{v^{i}\left( {{{uI}_{i}T_{e}} + {\Delta \; j_{i}T_{e}} + t_{i}} \right)}b_{{{mJ}_{i} + {\Delta \; m_{i}} - u},i}}}} + {b(t)}}}\; {{{with}\mspace{14mu} 0} \leq j < {I.}}} & (11) \end{matrix}$

Where j=Δm_(i)I_(i)+Δj_(i) with 0≦Δj_(i)<I_(i) and 0≦Δm_(i)≦J_(i).

Inter Symbol Interference

The observation vector x^(i)(t) arising from the antenna array at the instantt=m I_(i) T_(e)+jT_(e) involves, according to equation (10), the symbol b_(m,i) but also the symbols b_(m-u,i) where u is a relative integer belonging to the interval [r_(min,i)−L_(i), r_(max,i)+L_(i)], a phenomenon which is better known by the name Inter Symbol Interference (ISI). We denote by L_(c,i) this number of symbols participating in the ISI and we bound the interval of values taken by it. Thus, still according to equation (10), if the intersection of the intervals [r_(pi)−L_(i), r_(pi)+L_(i)] is not empty, then we have L_(c,i)=|r_(max,i)−r_(min,i)|+2L_(i)+1. Therefore, when r_(max,i)=r_(min,i), that is to say when, more concretely, all the multi-paths are correlated, the lower bound of L_(c,i) is attained and equals L_(c,i)=2L_(i)+1. This case is also conveyed mathematically by

${{{\max\limits_{p}\left\{ \tau_{pi} \right\}} - {\min\limits_{p}\left\{ \tau_{pi} \right\}}}} < {T_{i}.}$

On the other hand, if the intersection of said intervals is empty, and if, where relevant, all the intervals [r_(pi)−L_(i), r_(pi)+L_(i)] are disjoint, then we have L_(c,i)=P_(i)x(2L_(i)+1), thereby constituting an upper bound to the set of values capable of being taken by L_(c,i). The latter typical case corresponds concretely to the case of multi-paths all of which are decorrelated pairwise, which mathematically can also be written ∀ k≠l, |r_(ki)−r_(li)|>2L_(i), a condition obtained as soon as |τ_(ki)−τ_(li)|>(2L_(i)+1)T_(i). To summarize, the quantity L_(c,i) generally satisfies the following bracketing:

2L _(i)+1≦L _(c,i) ≦P _(i) X(2L _(i)+1)  (12).

Expression (10) can then be rewritten in the following manner, where now only the L_(c,i) symbols b_(m-u,i) of interest appear:

$\begin{matrix} {{{.{x^{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e}} \right)}} = {\sum\limits_{l = 1}^{L_{c,i}}{{h^{i}\left( {{{n(l)}I_{i}T_{e}} + {jT}_{e}} \right)}b_{{m - {n{(l)}}},i}\mspace{14mu} {with}}}}{0 \leq j < {I_{i}.}}} & (13) \end{matrix}$

Where ∀1≦l≦L_(c,i), and r_(min,i)−L_(i)≦n(l)≦r_(min,i)+L_(i) and where:

$\begin{matrix} {{.{h^{i}(t)}} = {\sum\limits_{p = 1}^{P_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{{h_{Fi}\left( {t - \tau_{pi}} \right)}.}}}} & (14) \end{matrix}$

Equation (11) then becomes at t=m I T_(e)+jT_(e):

$\begin{matrix} {{x\left( {{m\; I\; T_{e}} + {jT}_{e}} \right)} = {{\sum\limits_{i = 1}^{M}{\sum\limits_{l = 1}^{L_{c,i}}{{h^{i}\begin{pmatrix} {{n(l)I_{i}T_{e}} +} \\ {{\Delta \; j_{i}T_{e}} + t_{i}} \end{pmatrix}}b_{{{mJ}_{i} + {\Delta \; m_{i}} - {n{(l)}}},i}}}} + {{b(t)}.}}} & (15) \end{matrix}$

with 0≦j<I

Where j=Δm_(i) I_(i)+Δj_(i) with 0≦Δj_(i)<I_(i) and 0≦Δm_(i)<J_(i).

Variant Implementations of the Method According to the Invention

The rest of the description comprises, by way of wholly nonlimiting illustration, two variant embodiments of the method according to the invention. In the different cases, the separation techniques are applied to signals consisting of the mixture of symbols originating from one and the same transmitter and different propagation channels and of the mixture of the symbols arising from the various transmitters.

The first variant implementation of the method consists first of all in estimating the symbols {b_(k,i)} (symbol on carrier of the i^(th) transmitter), then in deducing the baseband symbols of the i^(th) transmitter {a_(k,i)} after estimating the frequency f_(i) (frequency corresponding to transmitter i). For this purpose, the procedure executes a first step consisting notably in separating the signals of the various transmitters by an ICA technique and a second step consisting notably in extracting the symbols {b_(k,i)} of each transmitter on the basis of the separated signals arising from the 1^(st) step. The 3^(rd) step consists in estimating the carrier frequency f_(i) on the basis notably of the {b_(k,i)} then in deducing the {a_(k,i)}. The extractions are done by ICA (Independent Component Analysis) type procedures described for example in references [3] [4] [8] [10].

The different variant embodiments of the method according to the invention involve, for example, ICA separation procedures based on the following signal model:

$\begin{matrix} {u_{k} = {{{\sum\limits_{i = 1}^{L}{g_{i}s_{ik}}} + n_{k}} = {{G\; s_{k}} + {n_{k}.}}}} & (16) \end{matrix}$

where U_(k) is a vector of dimension M×1 received at the instant k, s_(ik) is the i^(th) component of the signal s_(k) at the instant k, n_(k) is the noise vector and G=[g₁ . . . g_(L)]. The objectives of ICA procedures are notably to extract the I=L components s_(ik) and to identify their signatures g_(i) on the basis of the observations U_(k). The number I=L of components must be less than or equal to the dimension M of the observation vector. The procedures described in references [3] [4] [8] use the statistics of order 2 and 4 of the observations u_(k).

The first step uses the order-2 statistics of the observations to obtain a new observation Z_(k) such that:

$\begin{matrix} {z_{k} = {{W_{1}u_{k}} = {{{\sum\limits_{i = 1}^{L}{{\overset{˘}{g}}_{i}s_{ik}}} + {\overset{\sim}{n}}_{k}} = {{\overset{˘}{G}s_{k}} + {{\overset{\sim}{n}}_{k}.}}}}} & (17) \end{matrix}$

where the signatures {hacek over (g)}_(i) (1≦i≦L) are orthogonal, {hacek over (G)}=[{hacek over (g)}₁ . . . {hacek over (g)}_(L)] and s_(k)=[s_(1k) . . . s_(Lk)]^(T).

The second step consists in identifying the orthogonal basis of the {hacek over (G)} from the order-4 statistics of the whitened observations Z_(k). Under these conditions, it is possible to extract the signals s_(k) by performing:

ŝ_(k)={hacek over (G)}^(#)W₁u_(k)  (18).

Where ŝ_(k) is the estimate of the signals s_(k) and where ^(#) is the pseudo-inversion operator defined by {hacek over (G)}^(#)=({hacek over (G)}^(H){hacek over (G)})⁻¹ {hacek over (G)}^(H).

The ICAR procedure [10] uses for its part solely the statistics of order 4 to identify the matrix G=[g₁ . . . g_(K)] of the signatures.

FIRST VARIANT EMBODIMENT MIMO Demodulation with an ICA Step for Separating the Transmitters

FIG. 7 shows diagrammatically a 1^(st) variant of the method comprising a first step of separating the transmitters by an ICA technique and a second step consisting notably in extracting the symbols of each transmitter by an ICA technique.

Each transmitter is composed of Q_(i) groups of correlated multi-paths. The multi-paths of the i^(th) transmitter whose delays satisfy |τ_(ki)−τ_(li)|<(2L_(i)+1) T_(i), are mutually correlated by satisfying: E[s(t−τ_(ki)) s(t−τ_(ij))*]≠0. The model of equation (5) becomes:

$\begin{matrix} {{x^{i}(t)} = {{\sum\limits_{q = 1}^{Q_{i}}{\sum\limits_{p = 1}^{P_{iq}}{\rho_{piq}{a\left( \theta_{piq} \right)}{s_{i}\left( {t - \tau_{piq}} \right)}}}} = {\sum\limits_{q = 1}^{Q_{i}}{A_{iq}\Omega_{iq}{{s_{i}\left( {t,{\underset{\_}{\tau}}_{iq}} \right)}.}}}}} & (19) \end{matrix}$

Where A_(iq)=[a(θ_(1iq)) . . . a(θ_(Piq,i,q))], Ω_(iq)=diag([ρP_(q,i,q)]) and s_(i)(t, τ _(iq))=[s_(i)(t−τ_(1iq)) . . . s_(i)(t−τ_(Pq,i,q))]^(T) with τ _(iq)=[τ_(1iq) . . . τ_(Pq,i,q)]^(T). Consequently, the signal x(t) may be written:

$\begin{matrix} {{x(t)} = {{\sum\limits_{i = 1}^{M}{\sum\limits_{q = 1}^{Q_{i}}{A_{iq}\Omega_{iq}{s_{i}\left( {t,{\underset{\_}{\tau}}_{iq}} \right)}}}} + {{b(t)}.}}} & (20) \end{matrix}$

By applying an ICA procedure to the signal x(t), the following signals and signatures are estimated at the output of the separator according to [6]:

Â=[â₁ . . . â_(k)]=[A₁₁U₁₁ . . . A_(iq)U_(iq) . . . A_(M,QM)U_(M,QM)]Π

$\begin{matrix} {{{and}\mspace{14mu} {\hat{s}(t)}} = {\begin{bmatrix} {{\hat{s}}_{1}(t)} \\ \vdots \\ {{\hat{s}}_{K}(t)} \end{bmatrix} = {{\Pi \begin{bmatrix} {V_{11}{s_{1}\left( {t,{\underset{\_}{\tau}}_{11}} \right)}} \\ \vdots \\ {V_{iq}{s_{i}\left( {t,{\underset{\_}{\tau}}_{iq}} \right)}} \\ \vdots \\ {V_{M,Q_{M}}{s_{M}\left( {t,{\underset{\_}{\tau}}_{M,Q_{M}}} \right)}} \end{bmatrix}}.}}} & (21) \end{matrix}$

Where

$K = {{\sum\limits_{i = 1}^{M}{\sum\limits_{q = 1}^{Q_{i}}{P_{{iq},}U_{iq}V_{iq}}}} = \Omega_{iq}}$

and V_(iq) E[S(t, τ _(iq)) S(t, τ _(iq))^(H)] V_(iq) ^(H)=I_(Piq). The matrix Π is a permutation matrix.

Separation of the

$\sum\limits_{i = 1}^{M}Q_{i}$

Groups of Correlated Multi-Paths

The decorrelated paths such that E[s_(i)(t−τ_(piq)) s_(i)(t−τ_(pi,q′))*]=0 are received on different pathways Ŝ_(k)(t) and Ŝ_(l)(t). The correlated paths where E[s_(i)(t-−τ_(piq)) s_(i)(t−τ_(p′iq))*]≠0 are present on P_(Qi) pathways Ŝ_(k)(t). The different transmitters i and j satisfy E[s_(i)(t) s_(j)(t)*]=0 and their multi-paths are present on different pathways Ŝ_(k)(t). In the 1^(st) step of this variant, the method separates the multi-paths that are decorrelated on different pathways and makes it possible to identify the

$\sum\limits_{i = 1}^{M}Q_{i}$

groups of correlated multi-paths of all the transmitters indexed by “i”. By taking the outputs k and l of the separator, the following two assumptions can be tested:

$\begin{matrix} {H_{0}\text{:}\mspace{11mu} \left\{ {\begin{matrix} {{{\hat{s}}_{k}(t)} = {b_{k}(t)}} \\ {{{\hat{s}}_{l}(t)} = {b_{l}(t)}} \end{matrix}\mspace{14mu} {and}\mspace{14mu} H_{1}\text{:}\mspace{11mu} \left\{ \begin{matrix} {{{\hat{s}}_{k}(t)} = {{\alpha_{k}{s_{i}\left( {t - \tau_{ip}} \right)}} + {b_{k}(t)}}} \\ {{{\hat{s}}_{l}(t)} = {{\alpha_{l}{s_{i}\left( {t - \tau_{ip}} \right)}} + {{b_{l}(t)}.}}} \end{matrix} \right.} \right.} & (22) \end{matrix}$

where E[b_(k)(t) b_(l)(t−τ)*]=0 whatever the value of τ. Thus, under the assumption H₀ there are no multi-paths common to the two outputs k and l and under the assumption H₁ there is at least one for the i^(th) transmitter.

The test consists in determining whether the outputs Ŝ_(k)(t) and Ŝ_(l)(t−τ) are correlated for at least one of the values of τ satisfying |τ|<τ_(max). For this purpose, Gardner's test [2]-, which compares the following likelihood ratio with a threshold, is for example applied:

$\begin{matrix} {{{V_{kl}(\tau)} = {{- 2}K\; {\ln \left( {1 - \frac{{{{\hat{r}}_{kl}(\tau)}}^{2}}{{{\hat{r}}_{kk}(0)}{{\hat{r}}_{ll}(0)}}} \right)}\mspace{14mu} {with}}}{{\hat{r}}_{kl} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{{\hat{s}}_{k}(t)}{{{\hat{s}}_{l}\left( {t - \tau} \right)}^{\star}.}}}}}} & (23) \end{matrix}$

Where V_(kl)(τ)<η

assumption H₀ And V_(kl)(τ)≧η

assumption H₁

The threshold η is determined in [2] with respect to a chi-2 law with 2 degrees of freedom. One first of all seeks the outputs associated with the 1^(st) output by conducting the test for 2<l≦K and k=1. Then from the list of the outputs are removed all the outputs associated with the 1^(st) which will constitute the 1^(st) group with q=1. The same series of tests is repeated with the other outputs not correlated with the 1^(st) output so as to constitute the 2^(nd) group. This operation is performed until the last group where, finally, there are no more output pathways. As output from the sort operation the following is ultimately obtained:

Â _(n) =A _(iq) U _(iq) and Ŝ _(n)(t)=V _(iq) s _(i)(t,τ _(iq)) for (1≦q≦Q _(i)) and (1≦i≦M)  (24)

We therefore obtain

$K_{M} = {\sum\limits_{i = 1}^{M}Q_{i}}$

matrix-vector pairs (Â_(n), Ŝ_(n)(t)) associated with the q^(th) group of correlated multi-paths of the i^(th) transmitter.

The association between the index “n” and the pair of indices (i,q) is done on completion of the demodulation on each of the vectors Ŝ_(n)(t). The vector Ŝ_(n)(t) is a linear combination of P_(i) correlated multi-paths of the i^(th) transmitter.

Incidence and Delay Estimation of Each Group of Multi-Paths.

According to an optional step, the incidences θ_(p,i,q) (corresponding to the p^(th) multi-path, of the q^(th) group of correlated multi-paths of the i^(th) transmitter) are determined on the basis of the Â_(n) of equation (24) by applying for example the MUSIC algorithm [1] to the matrix Â_(n) Â_(n) ^(H). The matrices A_(n)=A_(iq)=[a(θ_(1,i,q)) . . . a(θ_(Piq,i,q))] are deduced from the goniometries thus obtained. Given that x_(n)(t)=Â_(n)Ŝ_(n)(t)=A_(iq)Ω_(iq) s_(i)(t, τ _(iq)), we deduce s_(i)i(t, τ _(iq)) to within a diagonal matrix by performing Ŝ_(i)(t,τ _(iq))=A_(iq) ^(#)X_(n)(t). As the elements of the Ŝ_(i)(t,τ _(iq)) are composed of the signals s_(i)(t−τ_(p,i,q)), the delays τ_(p,i,q−τ) _(1,i,l) are determined by maximizing the criteria c(τ)=|Ŝ_(i) ^(pq)(t−τ)−Ŝ_(i) ¹¹(t−τ)|² where Ŝ_(i)(t) is the p^(th) component of Ŝ_(i)(t, τ _(iq)).

Demodulation of the Vectors Ŝ_(n)(t) Arising from the 1^(st) ICA Step

The output vector Ŝ_(n)(t) arising from the 1^(st) step of separating the groups of correlated multi-paths is associated with the i^(th) transmitter and q^(th) multi-path group by satisfying: Ŝ_(n)(t)=V_(iq) s_(i)(t, τ _(iq)). The output Ŝ_(n)(t) is associated with a single transmitter with I_(i) samples per symbol that can be determined by a cyclic detection technique on Ŝ_(n)(t). As, according to equation (4), the vector s_(i)(t, τ _(iq)) satisfies:

$\begin{matrix} {\begin{matrix} {{s_{i}\left( {{{{mI}_{i}T_{e}} + {jT}_{e}},{\underset{\_}{\tau}}_{iq}} \right)} = \begin{bmatrix} {s_{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e} - \tau_{q,i,1}} \right)} \\ \vdots \\ {s_{i}\left( {{{mI}_{i}T_{e}} + {jT}_{e} - \tau_{q,i,P_{q\; \max}}} \right)} \end{bmatrix}} \\ {= {\sum\limits_{k = {- L_{i}}}^{L_{i}}{{h_{Fi}\left( {{{{kI}_{i}T_{e}} + {jT}_{e}},{\underset{\_}{\tau}}_{iq}} \right)}b_{{m - k},i}}}} \end{matrix}{{{for}\mspace{14mu} 0} \leq j < {I_{i}\mspace{14mu} {and}\mspace{14mu} {with}}}\text{}{{h_{Fi}\left( {{{{kI}_{i}T_{e}} + {j\; T_{e}}},{\underset{\_}{\tau}}_{iq}} \right)} = \begin{bmatrix} {h_{Fi}\left( {{{kI}_{i}T_{e}} + {jT}_{e} - \tau_{q,i,1}} \right)} \\ \vdots \\ {h_{Fi}\left( {{{kI}_{i}T_{e}} + {jT}_{e} - \tau_{q,i,P_{iq}}} \right)} \end{bmatrix}}} & (25) \end{matrix}$

it is possible to construct the following observation vector according to (24):

$\begin{matrix} {{{z_{n}\left( {{mI}_{i}T_{e}} \right)} = {\begin{bmatrix} {{\hat{s}}_{n}\left( {{mI}_{i}T_{e}} \right)} \\ {{\hat{s}}_{n}\left( {{{mI}_{i}T_{e}} + T_{e}} \right)} \\ \vdots \\ {{\hat{s}}_{n}\left( {{{mI}_{i}T_{e}} + {\left( {I - 1} \right)T_{e}}} \right)} \end{bmatrix} = {\sum\limits_{k = {- L_{i}}}^{L_{i}}{{h_{zi}(k)}b_{{m - k},i}}}}}{{{where}\mspace{14mu} {h_{zi}(k)}} = {\begin{bmatrix} h_{k,0}^{i} \\ h_{k,1}^{i} \\ \vdots \\ h_{k,{I_{i} - 1}}^{i} \end{bmatrix}.}}} & (26) \end{matrix}$

where h_(k,j)=V_(iq) h_(Fi)(k I_(i) T_(e)+jT_(e), τ _(iq))

This first variant embodiment can apply a procedure of ICA type [3] [4] [8] [10] to the observation vector Z_(n)(t) to estimate the 2L_(i)+1 symbol trains {b_(m-k,i)} indexed by “k”. The k^(th) output of the ICA procedure gives the symbol train {{circumflex over (b)}_(m,k,n)} associated with the channel vector ĥ_(zn,k), where {circumflex over (b)}_(m,k,n) is the estimate of the symbol b_(m−l,i). However, the estimated symbol trains {{circumflex over (b)}_(m,k,n)} arrive in a different order from that of the trains {b_(m−1,i)} while satisfying:

{circumflex over (b)} _(m,k,n)=ρ_(k) exp(jα _(k))b _(m−l,i) and ĥ _(z,n,k) =h _(zi)(l)  (27)

On completion of this step, the output {circumflex over (b)}_(m,k) _(max) _(,n) associated with the channel vector ĥ_(z,n,k,) _(max) of largest modulus is determined. As the symbol trains {b_(m−l,i)} are all of the same power, we write:

$\begin{matrix} {{\hat{b}}_{m,n} = {{\frac{{\hat{b}}_{m,k_{\max},n}}{\sqrt{E\left\lbrack {{\hat{b}}_{m,k_{\max},n}}^{2} \right\rbrack}}\mspace{14mu} {and}\mspace{14mu} {\hat{h}}_{z,n}} = {{\hat{h}}_{z,n,k_{\max}}.}}} & (28) \end{matrix}$

Where:

{circumflex over (b)} _(m,n)=exp(jα _(n))b _(m−l,i) and ĥ _(z,n) =h _(zi)(l)  (29)

On completion of this step, there remain Qi symbol trains {{circumflex over (b)}_(m,n)} associated with the same transmitter “i”.

Association of the Symbol Trains Dependent on the Same Transmitter

In this step, a factor L_(max) of maximum coherence of the symbols is chosen. It is therefore assumed that the temporal spreading of the transmitters does not exceed L_(max) T_(e). The step consists notably in associating the Qi outputs ({circumflex over (b)}_(m,n), ĥ_(z,n)) of each transmitter so as to extract therefrom a single one per transmitter. For this purpose, the sub-method performs a pairwise intercorrelation, of the outputs {circumflex over (b)}_(m,k) and {circumflex over (b)}_(m,l) by calculating the following criterion c_(k,l)(t):

$\begin{matrix} {{.{c_{k,l}(t)}} = {\frac{E\left\lbrack {{\hat{b}}_{m,k}{\hat{b}}_{{m - t},l}^{*}} \right\rbrack}{\sqrt{{E\left\lbrack {{\hat{b}}_{m,k}}^{2} \right\rbrack}{E\left\lbrack {{\hat{b}}_{{m - t},l}}^{2} \right\rbrack}}}.}} & (30) \end{matrix}$

When the function |c_(ij)(t)|>η for |t|<L_(max) is satisfied, the symbol trains {{circumflex over (b)}_(m,k)} and {{circumflex over (b)}_(m,l)} are associated with the same transmitter, where η is a threshold similar to the threshold usually used in Gardner's test [2].

Typically we take η=0.9 since 0<|c_(i,j)(t)|<1. When |c_(i,j)(t)|>η, the maximum of |c_(i,j)(t)| is at t=t_(max) for the k^(th) and l^(th) trains satisfying: {circumflex over (b)}_(m,k)={circumflex over (b)}_(m-t) _(max) _(,l). The algorithm for associating the outputs {circumflex over (b)}_(m,n) (1<n≦K) per transmitter is then composed of the following steps:

Step no A.1: Initialization: i=1, flag_(i)=0 and Φ_(i)={({circumflex over (b)}_(m,i), ĥ_(z,i))} tab_(i)={i} for (1≦i≦K) flag_(i), indicating whether the pair ({circumflex over (b)}_(m,i), ĥ_(z,i)) considered is already associated with a transmitter in one of the sets Φ_(i), tab_(i) corresponds to the index of the outputs of the set Φ_(i), Φ_(i) is the set of the outputs associated with one and the same transmitter, Step no A.2: Initialization to j=i+1. Step no A.3: Search for the maximum |c_(i,j)(t_(max))| at t=t_(max) for |t|<L_(max). Step no A.4: If |c_(i,j)(t_(max))|>η and flag_(j)=0 then flag_(j)=1, Φ_(i)−{Φ_(i)({circumflex over (b)}_(m,j), ĥ_(z,j))} and tab_(i)={ab_(i)j} Step no A.5: j=j+1 Step no A.6: If j<K then return to Step no A.3. Step no A.7: i=i+1 Step no A.8: If i<K then return to Step no A.2. Step no A.9: Determination of the M sets Φ_(i) where flag_(j)=0: the sets Φ′_(i) (1<i≦K) are obtained where Φ′_(i)={({circumflex over (b)}_(m,j), ĥ_(z,j)) such that jε tab_(i) }, which sets correspond to the pairs which are not associated with other sets. Step no A.10: Initialization: i=1. For each group we seek the most powerful output used to obtain the best symbol train at the output of the blind demodulator. Step no A.11: Search for the vector ĥ_(z,jmax) of maximum modulus in Φ′_(i). We will then write {circumflex over (b)}_(m,i)={circumflex over (b)}_(m,j) _(max) =exp(j α_(i)) b_(m,i) and ĥ_(z,i)=ĥ_(z,j) _(max) . i=i+1. Step no A.12: If i<M then return to Step no A.10.

On completion of this step, we obtain the symbols which are on carriers and which satisfy:

{circumflex over (b)} _(m,i)=exp(jα _(i))b _(m,i) with b _(m,i)=exp(j2πf _(i) mI _(i) T _(e))a _(m,i) for 1≦i≦M  (31)

Where f_(i) is the carrier frequency of the i^(th) transmitter.

Estimation of the Carrier Frequencies of Each of the Transmitters.

The objective of this step is to determine the baseband symbol trains {a_(m,i)} for 1I≦i≦M from the symbol trains {b_(m,i)}.

Accordingly, we estimate the carrier frequency f_(i) of the transmitter where the complex Z_(i)=exp(j2πf_(i)T_(e)) so as thereafter to deduce the symbols {a_(m,i)} from the symbols {{circumflex over (b)}_(m,i)} by performing according to (4):

â _(m,i) ={circumflex over (b)} _(m,i) exp(−j2πf _(i) mI _(i) T _(e))=b _(m) z _(i) ^(−mIi)  (32)

According to equations (4)(25)(26)(29), the vector ĥ_(z,i) with â_(m,i) satisfies:

$\begin{matrix} {{{\hat{h}}_{z,i} = {{h_{zi}(n)} = \begin{bmatrix} \begin{matrix} \begin{matrix} {z_{i}^{{nI}_{i}}{h_{i}\left( {{nI}_{i}T_{e}} \right)}} \\ {z_{i}^{{nI}_{i} + 1}{h_{i}\left( {{{nI}_{i}T_{e}} + T_{e}} \right)}} \end{matrix} \\ \vdots \end{matrix} \\ {z_{i}^{{nI}_{i} + {({I_{i} + 1})}}{h_{i}\left( {{{nI}_{i}T_{e}} + {\left( {I_{i} - 1} \right)T_{e}}} \right)}} \end{bmatrix}}}{with}{{h_{i}(t)} = {{V_{iq}\begin{bmatrix} \begin{matrix} {h_{i}\left( {t - \tau_{q,i,1}} \right)} \\ \vdots \end{matrix} \\ {h_{i}\left( {t - \tau_{q,i,P_{iq}}} \right)} \end{bmatrix}}.}}} & (33) \end{matrix}$

where h_(i)(t) is the waveform of the i^(th) transmitter.

Searching for f_(i) consists notably in maximizing the following criterion:

$\begin{matrix} {{{{Carrier}\left( f_{i} \right)} = {{{\hat{h}}_{z,i}^{H}{u_{i}\left( {\exp \left( {{- j}\; 2\; \pi \; f_{i}T_{e}} \right)} \right)}}}^{2}}{with}{{u_{i}(z)} = {{{c_{i}(z)} \otimes 1}\left( P_{iq} \right)}}{where}{{c_{i}(z)} = {{\begin{bmatrix} 1 \\ z \\ \vdots \\ z^{({I_{i} - 1})} \end{bmatrix}\mspace{14mu} {and}\mspace{14mu} 1(P)} = {\begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}.}}}} & (34) \end{matrix}$

Where 1(P) is a vector of dimension P×1. Once the frequency f_(i) is determined, the baseband symbol â_(m,i) is estimated by applying equation (32). However according to (31), the symbol â_(m,i) remains known to within a phase indeterminacy: α_(i)i.

Determination of the Phase Indeterminacy of the Symbols of the i^(th) Transmitter

Given that the estimated symbols satisfy according to (31):

â _(m,i)=exp(jα _(i))a _(m,I)  (35)

an objective of a step of the method is to estimate this phase α_(i). The first step will consist in identifying the constellation of the symbols a_(m,i) from among a database composed of the whole set of possible constellations. FIG. 6 shows an example of 8-QAM when α_(i)=0 and α_(i)≠0.

To identify the constellation and determine the phase, the method executes for example the following steps:

Step no B.1: Estimation of the positions of the states of the constellation (hatched points in FIG. 6) by searching for the maxima of the 2D histogram of the points M_(k)=(real(â_(k,i)), imag(â_(k,i))). For a constellation with M states, we obtain M pairs (û_(m), {circumflex over (v)}_(m)) for 1≦m≦M. Step no B.2: Determination of the type of constellation by comparing the position of the states (û_(m), {circumflex over (v)}_(m)) of the constellation of the {â_(k,i)}, with a database composed of the whole set of possible constellations. The closest constellation is composed of the states (u_(m), v_(m)) for 1≦m≦M. Step no B.3: Determination of the phase α_(i) by minimizing within the least squares sense the following system of equations:

û _(m)=cos(α_(i))u _(m)−sin(α_(i))v _(m) and {circumflex over (v)} _(m)=sin(α_(i))u _(m)+cos(α_(i))v _(m) for 1≦m≦M

To summarize; the first variant implementation of the method according to the invention comprises at least the following steps: Step no I.1: Application of an ICA separation type procedure [3] [4] [8] [10] to the observations x(t) (corresponding to the linear mixture of the multi-paths of each of the M transmitters) of equation (5) to obtain the vector output signal ŝ(t) of all the independent outputs which include the linear combinations of the multi-paths and the matrix Â of equation (21), the matrix being the mixture matrix of the independent outputs, Step no I.2: Extraction of the pair (ŝ(t), Â) from the pairs (Â_(n), ŝ_(n)(t)) of equation (24) associated with a group of correlated multi-paths of one of the transmitters: Identification of the independent outputs belonging to 2 groups of different multi-paths. Extraction of the groups of multi-paths, Step no I.3 (Option): For each pair (Â_(n), ŝ_(n)(t)) (group of multi-paths of a transmitter such that

$\left. {1 \leq n \leq {\sum\limits_{i = 1}^{M}Q_{i}}} \right)$

estimation of the incidences and delays of a group of correlated multi-paths of one of the transmitters, n is the index of a group, Step no I.5: n=1, for each group of multi-paths, Step no I.6: Determination of the symbol time T_(n) by applying a cyclic detection algorithm as in [2] [7] to ŝ_(n)(t), Step no I.7: Interpolation of the observations Ŝ_(n)(t) with I_(n) samples per symbol such that T_(n)=I_(n) T_(e), Step no I.8: Demodulation of the symbols {circumflex over (b)}_(m,n) of the n^(th) output Ŝ_(n)(t) associated with the channel vector ĥ_(z,n) of equation (28) (Paragraph 0). n=n+1 and return to step no I.8 if n<K (K: Number of independent outputs of ŝ(t)) we return to step I.5, Step no I.9: Grouping of the symbol trains {{circumflex over (b)}_(m,n)} associated with the same transmitter. For this purpose steps A.1 up to A.12 described previously are executed. The M pairs {{circumflex over (b)}_(m,i), ĥ_(z,i)} equation (31) for (1≦i≦M) are obtained at output, these pairs are associated with the most powerful output, Step no I.10: Estimation of the carrier frequencies fi of each transmitter so as to deduce the baseband symbol therefrom in the following manner: â_(m,i)={circumflex over (b)}_(m,i) exp(−j2πf_(i) m I_(i) T_(e)), Step no I.11: Determination of the phase rotation α_(i) of the symbol train {â_(m,i)} by applying steps B.1 to B.3 set out previously. Correction of the symbols â_(m,i) by performing for (1≦i≦M): a_(m,i)=exp(−j α_(i)) â_(m,i). The symbol trains {a_(m,i)} for (1≦i≦M) constitute the outputs of the MIMO demodulator of this sub-method,

SECOND VARIANT EMBODIMENT OF THE METHOD Direct MIMO Demodulation

The second variant embodiment consists in extracting the symbols {a_(k,i)} with a single step of ICA source separation by transforming the observation vector x(t).

FIG. 8 shows diagrammatically the steps of the second variant implementation of the method according to the invention. To simplify the notation, in this part the signal X(kT_(e)) is replaced by x(k). In this second variant implementation, the signal model of equation (15) is used. By assuming that there has been a phase of determining the symbol time T_(i)=I_(i) T_(e) for (1≦i≦M), it is possible to determine the greatest common multiple I of the I_(i)(1≦i≦M). The following spatio-temporal observation is then constructed:

$\begin{matrix} {\begin{matrix} {{{.z}\left( {{mI} + j} \right)} = \begin{bmatrix} \begin{matrix} \begin{matrix} {x\left( {{mI} + j} \right)} \\ {x\left( {{mI} + 1 + j} \right)} \end{matrix} \\ \vdots \end{matrix} \\ {x\left( {{mI} + \left( {I - 1} \right) + j} \right)} \end{bmatrix}} \\ {= {{\sum\limits_{i = 1}^{M}{\sum\limits_{l = 1}^{L_{c,i}}{{h_{z}^{i}\left( {{n(l)},{\Delta \; j_{i}}} \right)}b_{{{mJ}_{i} + {\Delta \; m_{i}} - {n{(l)}}},i}}}} + {b_{z}(t)}}} \end{matrix}{where}{{h_{z}^{i}\left( {n,\Delta} \right)} = {\begin{bmatrix} \begin{matrix} \begin{matrix} h_{n,\Delta}^{i} \\ h_{n,{\Delta + 1}}^{i} \end{matrix} \\ \vdots \end{matrix} \\ h_{n,{\Delta + I - 1}}^{i} \end{bmatrix}.}}} & (36) \end{matrix}$

with h_(n,j) ^(i)=h^(i)(nI_(i) T_(e)+jT_(e)+t_(i)), j=Δm_(i) I_(i)+Δj_(i) I_(i) J_(i)=I and b_(z)(mI)=[b(mI)^(T) . . . b(mI+(I−1))^(T)]^(T). Given that x(t) is of dimension N×1, the vector z(t) is then of dimension NI×1l. Each transmitter is associated with L_(c,i)XJ_(i) symbol trains {b_(mJ) _(i) _(+Δm) _(i) _(−(l),i)} for 0≦Δm_(i)<J_(i) and 1≦l≦L_(c,i). MIMO demodulation will therefore be possible when:

$\begin{matrix} {{\sum\limits_{i = 1}^{M}{L_{c,i} \times J_{i}}} \leq {{NI}.}} & (37) \end{matrix}$

If we are concerned solely with the M_(k) transmitters of symbol time I_(k), the model of equation (36) can be written at j=ΔI_(k) where 0≦Δ<J_(k):

$\begin{matrix} {{{.{z\left( {{mI} + {\Delta \; I_{k}}} \right)}} = {{\sum\limits_{i \in \Phi_{k}}{{H_{z}^{i}(0)}b_{{{mJ}_{k} + \Delta},i}}} + {\sum\limits_{l \in \Phi_{k}^{\bot}}{{H_{z}^{l}\left( {\Delta \; j_{l}} \right)}b_{{{mJ}_{l} + {\Delta \; m_{l}}},l}}} + {b_{z}(t)}}}\mspace{20mu} {where}\mspace{20mu} {{H_{z}^{i}(\Delta)} = \left\lbrack {{h_{z}^{i}\left( {{n(1)},\Delta} \right)}\mspace{14mu} \ldots \mspace{14mu} {h_{z}^{i}\left( {{n\left( L_{c,i} \right)},\Delta} \right)}} \right\rbrack}\mspace{20mu} {and}\mspace{20mu} {b_{{{mJ}_{i} + {\Delta \; m_{i}}},i} = {\begin{bmatrix} \begin{matrix} b_{{{mJ}_{i} + {\Delta \; m_{i}} - {n{(1)}}},i} \\ \vdots \end{matrix} \\ b_{{{mJ}_{i} + {\Delta \; m_{i}} - {n{(L_{c,i})}}},i} \end{bmatrix}.}}} & (38) \end{matrix}$

Where Δm_(i) I_(i)+Δj_(i)=Δ I_(k), Φ_(k) is the set of the indices of the transmitters of symbol time I_(k) and Φ_(k) ^(T) the set of the others. More simply still, it is possible to write:

$\begin{matrix} {{{.{z\left( {{mI} + {\Delta \; I_{k}}} \right)}} = {{H^{k}b_{{mJ}_{k} + \Delta}} + {{H^{k\bot}(\Delta)}b_{{mJ}_{k} + \Delta}^{\bot}} + {b_{z}(t)}}}{where}{H^{k} = {{\left\lbrack \mspace{14mu} {\ldots \mspace{14mu} {H_{z}^{i}(0)}\mspace{14mu} \ldots}\mspace{14mu} \right\rbrack \mspace{14mu} {and}\mspace{14mu} b_{{mJ}_{k} + \Delta}} = {{\begin{bmatrix} \begin{matrix} \vdots \\ b_{{{mJ}_{k} + \Delta},i} \end{matrix} \\ \vdots \end{bmatrix}\mspace{14mu} {for}\mspace{14mu} i} \in \Phi_{k}}}}{where}{{H^{k\bot}(\Delta)} = \left\lbrack \mspace{14mu} {\ldots \mspace{14mu} {H_{z}^{l}\left( {\Delta \; j_{l}} \right)}\mspace{14mu} \ldots}\mspace{14mu} \right\rbrack}\; {and}{b_{{mJ}_{k} + \Delta}^{\bot} = {{\begin{bmatrix} \vdots \\ b_{{{mJ}_{l} + {\Delta \; m_{l}}},l} \\ \vdots \end{bmatrix}\mspace{14mu} {for}\mspace{14mu} l} \in {\Phi_{k}^{\bot}.}}}} & (39) \end{matrix}$

Written globally we then have:

$\begin{matrix} {{{.{z\left( {{mI} + {\Delta \; I_{k}}} \right)}} = {{{H\left( {k,\Delta} \right)}b_{m,\Delta,k}} + {b_{z}(t)}}}{where}{{H\left( {k,\Delta} \right)} = {{\left\lbrack {H^{k}{H^{k\bot}(\Delta)}} \right\rbrack \mspace{14mu} {and}\mspace{14mu} b_{m,\Delta,k}} = {\begin{bmatrix} b_{{mJ}_{k} + \Delta} \\ b_{{mJ}_{k} + \Delta}^{\bot} \end{bmatrix}.}}}} & (40) \end{matrix}$

By applying a procedure for separating sources to the string of observations z(Δ I_(k)), z(l+Δ I_(k)), z(2I+αI_(k)) . . . , we obtain the symbol vectors be b_(m,Δ,k) and the channel matrix H(k,Δ) to within a diagonal matrix Ω(k,Δ) and a permutation matrix π(k,Δ), such that:

Ĥ(k,Δ)=H(k,Δ)Ω(k,Δ)π(k,Δ) and {circumflex over (b)} _(m,Δ,k)=π(k,Δ)⁻¹Ω(k,Δ)⁻¹ b _(m,Δ,k)  (41).

Given that the power of the symbols b_(k,i) is the same for all the transmitters we deduce that E[b_(m,Δk)b_(m,Δ,k) ^(H)]=I and that it is possible to put |Ω(k,Δ)|=I.

Extraction of the Transmitters with I_(k) Samples Per Symbol

By applying a source separation procedure to all the combs Δ associated with the observations z(m I+Δ I_(k)) for 0≦Δ<J_(k), it is possible to compare the results in Δ and Δ′ so as to identify the outputs associated with the transmitters with I_(k) samples per symbol. Under its conditions according to (41):

J(Δ′,Δ)=|Ĥ(k,Δ′)^(#) Ĥ(k,Δ)|=π(k,Δ′)⁻¹ |H(k,Δ′)^(#) H(k,Δ)|π(k,Δ)  (42)

Where ^(#) designates the pseudo-inverse. Noting according to (40) that the matrices H(k,Δ) for 0≦Δ<J_(k) are all composed of the channel matrix H^(k) of the transmitters with I_(k) samples per symbol, we obtain:

$\begin{matrix} {{J\left( {\Delta^{\prime},\Delta} \right)} = {{{\Pi \left( {k,\Delta^{\prime}} \right)}^{- 1}\begin{bmatrix} I_{L_{k}} & U \\ 0 & V \end{bmatrix}}{{\Pi \left( {k,\Delta} \right)}.}}} & (43) \end{matrix}$

where I_(K) is the identity matrix of dimension K×K

$L_{k} = {\sum\limits_{i \in \Phi_{k}}L_{c,i}}$

and U and V are matrices dependent on H^(kL)(Δ), H^(kL)(Δ′) and H^(k). Denoting by J_(ij)(Δ′,Δ) the ij^(th) element of the matrix J(Δ′,Δ), the associating of the outputs Ĥ(k,Δ) and Ĥ(k,Δ′) will consist in seeking the elements of the matrix J(Δ′,Δ) close to 1. The steps allowing the extraction of the symbol trains {b_(mJ) _(k) _(+Δ)} and the channel matrix H^(k) from the symbol trains {circumflex over (b)}_(m,Δ,k) and the channel matrices Ĥ(k,Δ) are the following: Step B-0: Δ=0 and Δ′=1. Initialization of the components of the vectors {circumflex over (b)}_(n) and of the matrix Ĥ^(k) to zero, Step B-1: Calculation of the matrix J(Δ′,Δ) of dimension P×P from Ĥ(k,Δ′) and Ĥ(k,Δ) according to (42), Step B-2: i=1 and j=1, Step B-4: If |J_(ij)(Δ′,Δ)−1|>η (close to 0) then the i^(th) output of Ĥ(k,Δ′) is associated with the j^(th) output of Ĥ(k,Δ). If |J_(ij)(Δ′,Δ)−1|≦η then jump to step B-8, Step B-5: Determination of the phase difference φ between pathways “i” and “j” which is the phase of:

${\exp \left( {j\; \phi} \right)} = \frac{{{\hat{h}}_{i}\left( {k,\Delta^{\prime}} \right)}^{H}{{\hat{h}}_{j}\left( {k,\Delta} \right)}}{\sqrt{\left( {{{\hat{h}}_{i}\left( {k,\Delta^{\prime}} \right)}^{H}{{\hat{h}}_{i}\left( {k,\Delta^{\prime}} \right)}} \right)\left( {{{\hat{h}}_{j}\left( {k,\Delta} \right)}^{H}{{\hat{h}}_{j}\left( {k,\Delta} \right)}} \right)}}$

where ĥ_(i)(k, Δ) is the i^(th) column of Ĥ(k,Δ) Step B-6: Construction of the symbol vectors {circumflex over (b)}_(n) where the j^(th) components satisfy: {circumflex over (b)}_(mJ) _(k) _(+Δ)(j)={circumflex over (b)}_(m,Δ,k)(j) and {circumflex over (b)}_(mJ) _(k) _(+Δ)(j)=exp(jφ) {circumflex over (b)}_(m,Δ′,k)(i). Step B-7: Filling in of the channel matrix Ĥ^(k) where j^(th) columns satisfy: Ĥ^(k)(j)=ĥ_(j)(k,Δ) and {circumflex over (b)}_(mJ) _(k) _(+Δ′)(j)=exp(jφ){circumflex over (b)}_(m,Δ′,k)(i), Step B-8:j=j+1 and return to step B-4 if j≦P, Step B-9: i=i+1 and return to step B-4 if i≦P. Step B-10: Elimination of the zero rows of the symbol trains {{circumflex over (b)}_(n)} and of the zero columns of the channel matrix Ĥ^(k).

At the end of this association, symbol trains {{circumflex over (b)}_(n)} and a channel matrix Ĥ^(k) associated with the transmitters with I_(k) samples per symbol are obtained. However they are still to within a permutation matrix the symbol trains {b_(mJ) _(k) _(+Δ)} and the channel matrix H^(k):

Ĥ ^(k) =H ^(k)π(k) and {circumflex over (b)} _(n)=π(k)⁻¹ b _(mJ) _(k) _(+Δ)  (44).

The objective of the following step is to separate the transmitters with I_(k) samples per symbol.

Separation of the Transmitters with I_(k) Samples Per Symbol

In this step, a factor L_(max) of maximum coherence of the symbols is chosen. It is therefore assumed that the temporal spreading of the transmitters does not exceed L_(max) T_(e). Accordingly by putting {circumflex over (b)}_(m,j) the i^(th) component of the vector {circumflex over (b)}_(m) and ĥ_(z,i) the i^(th) column of the vector Ĥ^(k), it suffices to apply steps A.1 to A.12 set out previously.

On completion of this step the estimated symbols are not in baseband while satisfying according to (4):

{circumflex over (b)} _(m,i)=exp(jα _(i))b _(m,i) with b _(m,i)=exp(j2πf _(i) mI _(k) T _(e))a _(m,i) for iεΦ_(k)  (45).

where f_(i) is the carrier frequency of the i^(th) transmitter.

Estimation of the Carrier Frequencies of Each of the Transmitters.

The objective of this step is to determine the baseband symbol trains {a_(m,i)} for iεΦ_(k) from the symbol trains {b_(m,i)}. We denote by ĥ_(z,i) a column of Ĥ^(k).

For this purpose, the carrier frequency f_(i) of the transmitter or the complex Z_(i)=exp(j2πf_(i)T_(e)) is estimated so as thereafter to deduce the symbols {a_(m,i)} from the symbols {{circumflex over (b)}_(m,i)} by performing according to (4):

â _(m,i) ={circumflex over (b)} _(m,i) exp(−j2πf _(i) mI _(k) T _(e))=b _(m) z _(i) ^(mIk)  (46)

According to (4)(14)(15)(36)(45) the vector ĥ_(z,i), with â_(m,i) satisfies:

$\begin{matrix} {{{\hat{h}}_{z,i} = {{h_{zi}(n)} = \begin{bmatrix} \begin{matrix} \begin{matrix} {z_{i}^{{nI}_{k}}{h_{i}\left( {{nI}_{k}T_{e}} \right)}} \\ {z_{i}^{{nI}_{k} + 1}{h_{i}\left( {{{nI}_{k}T_{e\;}} + T_{e}} \right)}} \end{matrix} \\ \vdots \end{matrix} \\ {z_{i}^{{nI}_{k} + {({I - 1})}}{h_{i}\left( {{{nI}_{k}T_{e}} + {\left( {I - 1} \right)T_{e}}} \right)}} \end{bmatrix}}}{with}{{h^{i}(t)} = {\sum\limits_{p = 1}^{P_{i}}{\rho_{pi}{a\left( \theta_{pi} \right)}{{h_{i}\left( {t - \tau_{pi}} \right)}.}}}}} & (47) \end{matrix}$

where h_(i)(t) is the waveform of the i^(th) transmitter.

Searching for f_(i) consists in maximizing the following criterion:

$\begin{matrix} {{{{Carrier}\left( f_{i\;} \right)} = {{{\hat{h}}_{z,i}^{H}{u_{i}\left( {\exp \left( {{- j}\; 2\; \pi \; f_{i\;}T_{e}} \right)} \right)}}}^{2}}{with}{{u_{i}(z)} = {{{c_{i}(z)} \otimes 1}\left( P_{iq} \right)}}{where}{{c_{i}(z)} = {{\begin{bmatrix} 1 \\ z \\ \vdots \\ z^{({I - 1})} \end{bmatrix}\mspace{14mu} {and}\mspace{14mu} 1(P)} = {\begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}.}}}} & (48) \end{matrix}$

Where 1(P) is a vector of dimension P×1. Once the frequency f_(i) is determined the baseband symbol â_(m,i) is estimated by applying equation (32). However according to (31), the symbol â_(m,i) remains known to within a phase indeterminacy: α_(i).

Determination of the Phase Indeterminacy of the Symbols of the i^(th) Transmitter

The phase α_(i) is determined by using for example equation (35) given previously.

To summarize; the second variant embodiment of the method according to the invention comprises for example the following steps:

Step no J.1: Determination of the symbol times T_(i) (1≦i≦K) by applying a cyclic detection algorithm as in [2] [7] to x(t), Step no J.2: Search for the greatest common multiple T of the T_(k) (1≦k≦K) and sampling at T_(e) of the observations x(t) such that T=I T_(e) and T_(k)=I_(k) T_(e) (1≦k≦K). Deduction of the indices T_(k)=I_(k) T_(e) (1≦k≦K), Step no J.3: Initialization k=1,

Step no J.4: Initialization to Δ=0,

Step no J.5: Application of an ICA procedure [3] [4] [8] [10] to the string of observations z(mI+ΔI_(k)) indexed by “m” of equation (38) to obtain the symbol trains {{circumflex over (b)}_(m,Δ,k)} and the channel matrix Ĥ(k,Δ), Step no J.6: Δ=Δ+1 and if Δ≦J_(k) return to step no J.5, Step no J.7: Extraction of the transmitters with I_(k) samples per symbol by applying steps B.0 to B.10 of paragraph 0: Obtaining of an estimate {{circumflex over (b)}_(n)} of the symbol trains {b_(mJ) _(k) _(+Δ)} as well as an estimate Ĥ^(k) of the channel matrix H^(k) of equation (40), Step no J.8 Extraction of {{circumflex over (b)}_(n)} from the trains of the symbols {{circumflex over (b)}_(n,i)} associated with the i^(th) transmitter such that iεΦ_(k). For this purpose it is necessary to apply steps A.1 up to A.12 of the paragraph association of the symbol trains dependent on the same transmitter. The M_(k) pairs {{circumflex over (b)}_(m,i), ĥ_(z,i)} of equation (36) for iεΦ_(k) are obtained as output, where ĥ_(z,i) is a column of the matrix Ĥ^(k) associated with the i^(th) transmitter, Step no J.9: Estimation of the carrier frequencies f_(i) of each transmitter so as to deduce therefrom the baseband symbol in the following manner: â_(m,i)={circumflex over (b)}_(m,i) exp(−j2πf_(i) m I_(k) T_(e)) (Paragraph (5)), Step no J.10: Determination of the phase rotation α_(i) of the symbol train {â_(m,i)} by applying steps B.1 to B.3 of paragraph (6). Correction of the symbols â_(m,i) by performing for (1≦i≦M): a_(m,i)=exp(−j α_(i)) â_(m.i). The symbol trains {a_(m,i)} for iεΦ_(k) constitute the outputs of the MIMO demodulator of this sub-method, Step no J.11: k=k+1 and if k≦K return to step no J.4.

REFERENCES

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1. A method of blind demodulation of signals arising from one or more transmitters, the signals including a mixture of symbols where the signals are received on a system comprising several receivers, comprising the steps of; separating the transmitters by using the temporal independence of the indexed symbol trains {a_(k−p,i)} where ρ is the index of a symbol train specific to a transmitter and the mutual independence of the transmitters indexed where i is the index of a transmitter.
 2. The method as claimed in claim 1, comprising a first step of separating the various transmitters on the basis of the observations received after having passed through a propagation channel and a second step consisting in extracting the best baseband symbol train {a_(k−p,i)} of each transmitter indexed by i on the basis of the separated signals of the first step, where k is the temporal index of a symbol train.
 3. The method as claimed in claim 1, comprising a step where the observation vector composed of the signals received after having passed through the propagation channel is transformed into a spatio-temporal vector and a step of separating the transmitters, executed on the spatio-temporal vector so as to jointly extract the baseband symbol trains {a_(k−p,i)} for each transmitter.
 4. The method as claimed in claim 3, comprising a step of determining the phase rotation of the symbol train. 